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Edited by: Asher Cohen, University of Jerusalem, Israel

Reviewed by: Asher Cohen, University of Jerusalem, Israel; Yoav Kessler, Ben-Gurion University of the Negev, Israel

*Correspondence: Attila Krajcsi, Department of Radiology, Harvard Medical School, Brigham and Women’s Hospital, Boston, MA 02115, USA. e-mail:

This article was submitted to Frontiers in Cognition, a specialty of Frontiers in Psychology.

This is an open-access article distributed under the terms of the

Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we compared artificial sign-value and place-value notations in simple numerical tasks. We found that, contrary to the dominant view, sign-value notation can be applied more easily than place-value notation for multi-power comparison and addition tasks. Our results are consistent with the popularity of sign-value notations that prevailed for centuries. To explain the notation effect, we propose a natural multi-power number representation based on the numerical representation of objects.

Numbers can be represented in many ways. Indo-Arabic numbers and Roman numerals are only two well-known examples, while dozens of other notations were invented throughout the history of human culture (Ifrah,

In a complex notational system, numbers are decomposed into multiples of powers. For example, a specific number could be denoted as the sum of 1s, 10s, 100s, etc. Different cultures applied different methods to utilize this power decomposition. Many methods used two common structures to denote numbers: sign-value and place-value notations (Table

Roman example: XXIII | Indo-Arabic example: 23 | |
---|---|---|

Noting the powers | ||

(e.g., 1, 10, 100 in a base 10 system) | X means ten | □_ (left position) means tens |

I means one | _□ (right position) means ones | |

Noting the quantity | ||

within a specific | ••• Means three | 3 Means three |

power | •• Means two | 2 Means two |

Understanding the role of notation in numerical processing relies on at least two factors: first, how multi-power numbers are represented mentally, and second, how the specific notation is transcoded to those mental representations. Regarding the first question, research in recent decades has revealed that humans may represent the very same number in different forms; however, there is no consensus on the nature of multi-digit number representation. In an initial model, McCloskey (

Turning to the second question, the role of transcoding in handling different number notations has been examined in only a handful of studies. The mental use of the aforementioned abstract multi-power number representation was tested with the utilization of Roman number notation (Gonzalez and Kolers,

It is not only cognitive scientists who find multi-power representation and number notation important. Mathematicians and historians also provide relevant considerations; however, statements about the efficiency of sign-value vs. place-value notations seem somewhat paradoxical. On one hand, there is a strong consensus in the literature that the place-value system is highly efficient (Cajori,

To summarize, the literature on number notations has diverse findings. While we have solid knowledge of multi-power number representation (Dehaene,

In the present study, we investigated the cognitive effect of number notations on mental number processing. More specifically, based on historical considerations (i.e., sign-value numbers were popular even when place-value alternatives were available) and computational considerations (e.g., Roman numbers can be used easily to make calculations), we addressed the simple but fundamental problem of whether place-value notation is more complex for human numerical processing, or if, in contrast, it is the sign-value notation that is more complex for human calculation as suggested by the majority of cognitive and mathematical literature (Cajori,

To overcome the methodological issues described above, two structurally comparable artificial number notations were designed, and participants solved simple multi-power comparison and addition tasks in the new notations. We hypothesized that, contrary to the conventional view, sign-value notation might be more appropriate for number processing than place-value notation. This hypothesis is based on the relative popularity of sign-value notation in the history of culture (Menninger,

In the first experiment, participants compared numbers in a new artificial number system. A comparison task was chosen as one of the simplest numerical tasks that also had high importance in ancient cultures (Ifrah,

To mitigate potentially unavoidable interference with the well-known Indo-Arabic numbers, the new notational systems were as different from our usual notations as possible. Base 4 systems with new characters were designed. All characters had similar vertical and horizontal extent and position, and all of them were visually complex: 0-Ł, 1-Ɵ, 2-Đ, 3-И, 4-Я, and 16-Ҹ (Figure

The two notational systems should differ only in their structure (sign-value vs. place-value), but not in other aspects. First, in contrast with some Roman numbers, such as IV or IX, the sign-value notational system used here did not include subtraction, because subtraction is a deviation from the simplest form of sign-value notation, and it is also a deviation from the place-value structure applied here. Second, the number of symbols a user learns and the length of the numbers should be approximately equal in the two notations. The number of different symbols and the length of the numbers are determined by the base of the number system and the largest power that can be expressed in the task (see Table ^{3}−1, i.e., 26). The length of the numbers in the two notations are closer to each other if the base number and the largest power are small, although it is difficult to perfectly control this aspect of the number systems. Considering these viewpoints, in the present experiment, the base 4 system and three powers were chosen. Thus, the number of symbols was 3 in the sign-value system and 4 in the place-value system. The average length of a number was 4.57 digits in the sign-value notation, and 2.71 in the place-value system.

Sign-value system | Place-value system | |
---|---|---|

Number of symbols | Largest power: e.g., in a base 10 system, if the largest power is hundred, i.e., 10^{3}, three different symbols are used |
Base of the system: e.g., the digits from 0 to 9 in base 10 |

Length of a number | Determined by the base and the power together: e.g., for all powers, within the power, the length can be between 0 and the base number | Largest power: e.g., in a base 10 system if the largest power is hundred, i.e., 10^{2}, then the longest number can consist of three digits |

The comparison tasks can be solved by incorrect strategies. For example, one can count the number of symbols in a sign-value notation; utilizing this strategy to compare III vs. XX in Roman notation, one would produce an erroneous solution, stating that III is larger than XX, as the first number includes three symbols, while the second number includes only two. Notably, these alternative incorrect strategies give a correct answer for most of the specific number pairs. For example, applying the above mentioned number of symbols, strategy would state that Roman XXX is larger than XX, which in fact is true, although not because XXX includes more digits than XX; rather, the sum of the values in XXX give a larger number than the values of X plus X.

We identified seven strategies that could offer an alternative solution in the comparison task. Some of them might sound bizarre, but we wanted to be confident that the occasionally confused participants were not using any alternative strategies that could fill the gap made by their uncertainty. Some strategies could be applied in only one notation. The seven strategies were:

As mentioned above, in many cases these incorrect strategies will give the correct solution; thus, only the number pairs in which a specific incorrect strategy gives an erroneous result are informative. It would be appealing to find number pair stimuli in which only one incorrect strategy would propose the wrong solution, and the correct strategy, along with all the other wrong strategies, would give the correct answer, as failing these trials would reveal that the specific wrong strategy was used. Unfortunately, this is impossible in some cases. For example, in sign-value notation, when the number of digits would suggest a wrong solution (Strategy 2), the utilization of the “not largest differing power” strategy (Strategy 5) also offers an incorrect response. There are no cases where Strategy 2 gives the wrong solution and Strategy 5 gives the correct result. Thus, whenever it was not possible to choose a number pair where only a single incorrect strategy would propose the wrong answer, we chose an “overlapping” number pair stimulus, in which only two wrong strategies support the wrong result: one is the critical one that we wish to test, and the other is a strategy that could be tested by itself (i.e., there are other number pairs in which the latter incorrect strategy is the only strategy that gives an incorrect result).

Thirty Hungarian undergraduate students from Eötvös Loránd University participated in the study for partial course credit. All participants had normal or corrected to normal vision. The data of 24 subjects were analyzed (two males, age range from 20 to 27) after excluding six participants with a higher than 50% error rates in any of the tested incorrect strategies (see incorrect strategies above and the procedure below). These participants were excluded to ensure that all of the remaining participants understood the structure and logic of both the sign-value and place-value notations. Among the excluded participants, five subjects applied a strategy that counted the number of symbols instead of adding up their values in sign-value notation. Another participant added up the values of the digits in place-value notation (i.e., the place-value numbers were handled as if they were sign-value numbers).

In a comparison task, two multi-power numbers were visible on the left and right sides of the screen, and participants chose the larger number (see Figure

The presented numbers were between 1 and 63; i.e., numbers that have a maximum of three powers in a base 4 system. The numbers were presented in white against black background. The stimuli were visible until the response button was pressed. A blank screen appeared for 500 ms between trials.

The tasks were presented in two blocks: sign-value and place-value notation blocks. In a block, first, participants learned the symbols; an instruction introduced the new symbols used in the notation. Then, the participants saw one symbol at a time in the middle of the screen and had to press the associated response button. After the response, auditory feedback was given depending on whether the response was correct or not (higher and lower beeps). All new digits were presented twice in a block. The participant performed this practice phase until a block was completed without any error and at least four blocks were accomplished. For most of the participants four blocks of practice was enough to identify the symbols accurately.

Second, the multi-power notation was introduced. To ensure that participants understood that the incorrect strategies described above were erroneous, practice trials tested whether the participants applied these critical strategies. All possible incorrect strategies of the notation were tested with five trials. After each erroneous practice trial, the notation was explained by the experimenter. The trials in the practice phase were randomized.

Finally, participants completed the comparison task. To ensure that participants continued to avoid incorrect strategies, test trials continuously monitored the use of possible wrong strategies while solving the comparison tasks. The incorrect strategies could be verified only with these test trials, as the main trials included numbers that could be solved correctly with any of the incorrect strategies (i.e., all incorrect strategies gave correct solutions to the main trials). An incorrect strategy was followed if more than two errors out of five trials were made in that strategy type.

Former studies on multi-digit Indo-Arabic number comparison have found that (a) the number of digits are used as a shortcut for a decision (i.e., longer numbers are always larger in a place-value system; Hinrichs et al.,

Sign-value number | Place-value numbers | |
---|---|---|

One leading zero | ҸЯЯƟƟƟ ЯЯƟƟƟ | ĐИ ƟĐИ |

Two leading zeros | ƟƟƟ ҸЯЯƟƟƟ | ƟĐИ И |

Difference in 16s | ҸЯЯƟƟƟ ҸҸЯЯƟƟƟ | ƟĐИ ĐĐИ |

Difference in 4s | ҸЯƟƟƟ ҸЯЯƟƟƟ | ƟŁИ ƟĐИ |

Difference in 1s | ҸЯЯƟƟ ҸЯЯƟƟƟ | ƟĐИ ƟĐƟ |

The full factorial within-subjects design included notation (sign-value and place-value) and the number difference type with five levels, as described above. Each cell of the design included 15 trials. The stimuli were presented in two notation blocks and the order of the notation was counterbalanced across subjects. In a block, the order of the trials with the number difference conditions and the incorrect strategy trials were randomized. The specific number pairs presented were generated online with the appropriate constrains of the conditions: all stimuli were chosen randomly from the set of number pairs that satisfy the appropriate constrains. Presentation of the stimuli and measurement of RT were managed by PsychoPy software, version 1.61 (Peirce,

Error rates and response latencies were analyzed with a 2 (notation: sign- vs. place-value) × 5 (number difference: one leading zero vs. two leading zeros vs. difference in 16s vs. in 4s vs. in 1s) × 2 (order of notation: sign-value notation first vs. place-value notation first) ANOVA with notation and number difference as within-subjects and order of notation as between-subjects factor. Order of notation neither had a main effect nor interacted with any other factors. We found that comparison in place-value notation was more erroneous and slower than comparison in sign-value notation;

Further analysis explored the presence of comparison strategies formerly observed in multi-digit Indo-Arabic comparison tasks (Hinrichs et al.,

These behavioral data are consistent with the formerly known multi-digit Indo-Arabic number processing. First, response latencies show a relatively fast solution for trials with leading zeros (Hinrichs et al.,

To investigate the performance improvement over time, the trials were grouped into four blocks. A 2 (notation) × 4 (blocks) repeated measures ANOVA on error rates revealed only a main effect of notation, but not a main effect of blocks or interaction. A similar 2 × 4 ANOVA on the response latencies (Figure

The comparison tasks, in a strict sense, could have been solved by ordering the symbols without considering their values. To rule out this potential problem, we utilized the more complex task of addition. Moreover, in the comparison task, participants might have misunderstood the base 4 system and could have interpreted the multi-power numbers as base 10 numbers, as both interpretations implied the same solution. However, in the addition task, misunderstanding the base would cause a carry error, for example, 2 + 3 is 5 in a base 10 system, but the result is (1)(1) in a base 4 notation. Third, addition was also a vital procedure in ancient times, as complex notations were frequently applied for addition, such as for summing assets or taxes (Ifrah,

Twenty-two Hungarian undergraduate students from Eötvös Loránd University participated in the study for partial course credit. Nineteen subjects were analyzed (four males, age range from 20 to 23) after excluding participants with more than 20% overall error rate. In contrast with the comparison task, no incorrect strategies test trials were needed in the addition task, as the aforementioned incorrect strategies would result in wrong solutions in the addition tasks. Consequently, exclusion of the participants was not based on incorrect strategy test trials, but on the overall performance in the addition task.

The same artificial notational system was applied as in the previous experiment. In a trial, a multi-power addition was visible in the middle of the screen, and a proposed solution appeared at the bottom of the screen (see Figure

To compare the present artificial number-learning paradigm with the formerly known multi-digit Indo-Arabic addition, two manipulations of the stimuli were applied. To investigate serial power-by-power processing (Deschuyteneer et al.,

The tasks were presented in two blocks: sign-value and place-value notation. In each block, participants first learned the new symbols with the same procedure as in the first experiment, then practiced the addition in the new notation. To ensure that participants understood the task, the following types of additions were practiced: (1) in sign-value notation, two one-digit addends; (2) in place-value notation, two one-digit addends with no carry; (3) in both notations, multi-power addition without carry; (4) in both notations, multi-power addition with one carry; and (5) with two carries. After each erroneous practice trial, the rules of addition were explained by the experimenter.

In the main part of the experiment, half of the trials showed a correct result, and the other half showed an incorrect sum. In a notation block, 120 trials were presented. In both notations, a factorial design included erroneous power and carry factors, with 20 trials in each erroneous condition, 60 trials in the correct result condition, and 30 trials in each carrying condition (see the detailed distribution of the trials in the design cells in Table

Erroneous result |
Correct result |
|||
---|---|---|---|---|

Error on 16s | Error on 4s | Error on 1s | No error | |

No carry-over | 5 | 5 | 5 | 15 |

Carry-over from 1s | 5 | 5 | 5 | 15 |

Carry-over from 4s | 5 | 5 | 5 | 15 |

Carry-over from 1 to 4s | 5 | 5 | 5 | 15 |

A 2 (notation: sign-value vs. place-value notation) × 4 (erroneous power: no error vs. error in 16s vs. error in 4s vs. error in 1s) × 4 (carry: no carries vs. carry from 1s vs. carry from 4s vs. carry from 1s to 4s) repeated measures ANOVA was applied to analyze error rates and median response latencies of correct responses. We found that, consistent with the comparison task, addition with place-value notation was slower than addition with sign-value notation [

Further analysis explored the effects formerly known from multi-digit Indo-Arabic addition (Deschuyteneer et al.,

The previous ANOVA on response latencies (Figure

This more detailed analysis reflects a similar processing in the addition task than was observed previously with multi-digit Indo-Arabic notation (Deschuyteneer et al.,

To test the effect of the order of presentation a 2 (notation) × 2 (notation order) ANOVA was run with notation as a within-subject and order of notation as a between-subject factor. The ANOVA on error rates did not show a main effect neither for the notation nor for the order of the notation, while the interaction was significant,

To investigate the performance improvement over time, trials were grouped into four blocks (see Figure

Adults can compare and add numbers better in sign-value notation than in place-value notation. One possible explanation for this phenomenon is that adults have extensive experience with place-value Indo-Arabic numerals, and this former knowledge about a place-value system could cause interference in the newly acquired place-value system, while the sign-value system is untouched by such an influence. To investigate this possibility, the comparison task was solved by preschool children who have less experience with the place-value system and their performance shows higher variance.

The previous results could have been attributed to the specific symbol-value assignments. For example, the symbol Ɵ meant 1, although the symbol itself may remind some participants of the digit 0. This interference might cause an artifact, and the results described above could be attributed to this uncontrolled factor. To control for this possible artifact the symbol-value assignment was randomized for every participant. If the previous results are the consequence of a specific symbol-value assignment artifact, then the effects should disappear or at least diminish in the random symbol-value assignment version.

Forty-five preschool children participated in the study; 24 girls and 21 boys; mean age 6–5, range 5–8 to 7–5.

The same comparison task was used as in Experiment 1, with the following modifications. Each cell of the design included 10 trials instead of 15 trials, to shorten the length of the experiment. The same practice trials that were applied in the adult version were repeated twice to support the learning process. The same symbols were used as in Experiment 1; however, the values of those symbols were randomized for every participant. For example, the symbol Ɵ could mean 1 for some participants, 2 for some others, etc.

Understanding the new artificial notation could be influenced by former knowledge of other place-value and sign-value number notations. To control for the effect of the already learned Indo-Arabic and Roman numbers, a number reading task was given. Children had to read (a) single-digit Indo-Arabic, (b) multi-digit Indo-Arabic, and (c) Roman numbers. The numbers were presented in the middle of the screen, and children read the number out load. In the single-digit Indo-Arabic task all digits from 1 to 9 (0 excluded) were presented. In the multi-digit Indo-Arabic task, 10 numbers from 11 to 29 were presented; five even and five odd numbers, and five numbers from the teens and five from the twenties. In the Roman number task, all numbers from 1 to 9 were presented.

Understanding of the notations was measured with the error rates in the incorrect strategy test trials (see the ^{2}(1,

To control for former experiences with written number notations, knowledge about single- and multi-digit Indo-Arabic and Roman numbers were measured. Preschool children were confident with the Indo-Arabic numerals, making only 4% errors on average in reading single-digit Indo-Arabic numbers, and making 22% errors with multi-digit Indo-Arabic numbers. Roman numbers were not known by these children, as reflected in the 85% error rate. While preschool children have definitely less experience with Indo-Arabic numbers than adults, they still have some knowledge about the multi-digit place-value notation. Therefore it is possible that this cultural influence might already have an effect on the artificial number notation paradigm. To test this hypothesis, the error rate of multi-digit Indo-Arabic number reading was correlated with the error rate of both the sign-value and the place-value comparison tasks. If knowledge about multi-digit Indo-Arabic numbers causes bias on the new number notation learning, then a positive correlation with place-value and a negative correlation with sign-value notation is expected. Multi-digit Indo-Arabic number reading error rates correlated positively with both sign-value comparison error rate,

For further analysis, only the data of the children who understood both sign-value and place-value notations, verified by the incorrect strategy test trials, were used. The remaining nine children were two girls and seven boys, mean age 6–8, range 6–1 to 7–4. Although the children who understood both notations were somewhat older than those who had difficulties with them (6–8 vs. 6–5 years), the difference was not significant,

Error rates and median response latencies of the correct responses were analyzed with a 2 (notation: sign- vs. place-value) × 5 (number difference: one leading zero vs. two leading zeros vs. difference in 16s vs. in 4s vs. in 1s) repeated measures ANOVA. While error rate did not differ between the two notations, place-value notation comparisons were slower to solve than sign-value tasks with marginal significance,

As in Experiment 1, the comparison strategies were also investigated (Figure

The error rate and response latency patterns reflect the same strategies that were observed in adults in Experiment 1 and in former studies (Hinrichs et al.,

The specific symbol-value assignment was randomized in this experiment to control for the effect of specific symbols that might cause interference with other known symbols. The pattern of the errors and response latencies are basically the same as found in Experiment 1, suggesting that the effects described above can not be attributed to the specific symbols and their assignments utilized in our study.

To summarize, for preschool children, sign-value notation is easier to use than place-value notation in multi-power comparison tasks. This result did not depend on former experience with other written number notations. Finally, preschool children used the same strategies adults did.

It is possible that the notation effect is based on some non-essential properties of the notation system, rather than on the logic of the notation. Since the number notations display the powers in an ordered sequence (i.e., largest powers are on the left), in the case of sign-value notation it is not necessary to learn the symbols as strictly as in place-value notation, because the position of the symbol also gives information about the value of the symbol (e.g., the leftmost symbols should be 16s). This feature of the sign-value notation might contribute to the advantage of sign-value notation, or according to a more extreme scenario, it is possible that the whole notation effect is simply an artifact of the position information in sign-value notation.

Additionally, in the previous experiments base 4 systems with 3 powers were used. This means that while participants had to learn three symbols in sign-value notation, they had to learn four symbols in place-value notation. Again, sign-value notations had an advantage over place-value notations in the previous experiments.

To test the possible effect of position information and amount of symbols to learn on notation effect, we designed a follow-up experiment in which position information and the number of symbols were controlled.

Eighteen Hungarian undergraduate students from Eötvös Loránd University participated in the study for partial course credit. All participants had normal or corrected to normal vision. The data of 16 subjects were analyzed (two males, age range from 19 to 26) after excluding two participants with a higher than 50% error rates in any of the tested incorrect strategies (see incorrect strategies above and the procedure below).

The same stimuli were used and the same procedure was followed as in Experiment 1 with the following modifications.

To control for the number of symbols to be learned even more strictly, base 3 number system (instead of base 4 in the previous experiments) with three-power numbers were utilized. In this base 3 system, the powers could be 1, 3, and 9. Thus, the same number of symbols should be learned both in the sign-value notation (symbols for 1, 3, and 9) and in the place-value notation (symbols for 0, 1, and 2). The assignment of symbols and values were randomized as in Experiment 3 to ensure that the notation effect is not the result of some specific symbol processing.

To test the effect of position information in the sign-value notation, the position information should be removed. In the sign-value notation the position information cannot be used when some of the powers are missing in a number (which would be denoted by zeros in a place-value notation), thus, for example after 9s the participants cannot be sure whether the next symbols denote 3 or 1. Leading zeros should not be used as test stimuli, because leading zeros in place-value notation are processed by length shortcut as revealed in the first experiment. Thus, test number pairs had a x0x and xx0 power structure, in which x could be any non-zero power, and the largest powers were equal, so thus the task could be solved only on the medium power. In these tasks participants cannot rely on position information, because in the two numbers after the 9s different symbols show up, and to compare the powers, the participant must recall the value of that digit.

The new test condition was added to the five conditions applied in Experiment 1: (a) one leading zero, (b) two leading zeros, (c) difference in 9s, (d) difference in 3s, and (e) difference in 1s.

Error rates and median response latencies of the correct responses were analyzed with a 2 (notation: sign- vs. place-value) × 6 (number difference: one leading zero vs. two leading zeros vs. difference in 16s vs. in 4s vs. in 1s vs. the new test condition, x0x and xx0 pairs) × 2 (order of notation: sign-value notation first vs. place-value notation first) ANOVA with notation and number difference as within-subject and order of notation as between-subject factors (see Figure

The notation effect did not disappear in the “x0x and xx0,” in which the position information was unavailable for the participants in the sign-value notation. Thus, it is improbable that the notation effect would be solely the artifact of this extra position information in sign-value notation. To summarize, stricter control for the number of symbols to learn and the position information in the sign-value notation does not remove the notation effect.

In the present study, participants could more easily compare and add new base 4 (or base 3) artificial multi-power numbers in sign-value notation than in place-value notation, the result of which works against the conventional view of the overall superiority of place-value system. This finding cannot be the result of former experience with place-value Indo-Arabic system causing interference with the new artificial notations, since preschool children showed the same notation effect in the comparison task independently of their former Indo-Arabic number experiences. The relative advantage of sign-value notation is consistent with the popularity of sign-value notation in the history of culture (Ifrah,

Why was sign-value notation easier to process than place-value notation in simple numerical tasks? Representing multi-power numbers requires a special representation, although its nature is debated. According to former models, this representation is assumed to be in a verbal form (Dehaene et al.,

Because none of the three former number representation models accounts for the notation effect, we propose here an alternative model, termed the natural multi-power number representation, based on the numerical representation of objects and groups. This number representation might represent numbers as specific number of objects, in which some type of objects may represent items, while other types of objects can represent groups or higher powers (see the middle column in Figure

To summarize, the analogous structure of natural multi-power number representation and sign-value notation might make transcoding fast, while processing the place-value notation with a divergent structure requires more abstraction and causes difficulties. It is important to stress that we do not imply that the abstract code or the triple code model with the Arabic visual number form and verbal number representation would be invalid; rather, we propose a new complementary number representation that can be seen as the base of representing exact multi-power numbers. The hypothesis of natural multi-power number representation cannot be constructed from our results in a strict sense, as the experiment was not designed to explore detailed representational and transcoding issues; rather, the hypothesized representation seems a reasonable possibility in light of the theoretical consideration and in light of our result conflicting with previous assumptions about the superiority of place-value notations.

We highlight that only simple numerical tasks were tested in the present study, and all the presented artificial number systems were base 4 or 3 systems with maximum three powers in use. It is not entirely known how generalizable our results are. Changing the base or the power of the system changes the number of symbols a user should learn and changes the average length of the numbers. Both of these factors might influence the processing speed and the error rates. While in the present study we controlled for these factors to investigate only the effect of the structures of the notations, for example, a base 10 system with values of different order of magnitudes might be processed differently.

The data presented herein might explain why sign-value notation was popular for centuries, even when alternative place-value notation was available (Ifrah,

To summarize, our results reveal that if parameters other than the notation itself are controlled, then sign-value notation is easier to apply than place-value notation in simple numerical tasks. We propose that this notation effect can be explained with a natural multi-power number representation based on object representation. These results highlight the elementary role of number notations in number representation and imply that the effects originating from the number representation and the effects originating from the notation of numbers should be distinguished.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by Fidelity Foundation, Hungarian National Research Fund (PD 76403), and János Bolyai Research Scholarship of the Hungarian Academy of Sciences. We thank József Fiser for his comments on an earlier version of the manuscript.